To analyze the variation of the Coulomb force with respect to the inverse square of the distance between two charges, we can start by recalling Coulomb's Law. This law states that the force (F) between two point charges is directly proportional to the product of the magnitudes of the charges (q1 and q2) and inversely proportional to the square of the distance (r) between them. Mathematically, it can be expressed as:
F = k * (|q1 * q2| / r^2)
Where k is Coulomb's constant, approximately equal to 8.99 x 10^9 N m²/C². In our scenario, we will look at two pairs of charges: (1 microcoulomb and 2 microcoulombs) and (2 microcoulombs and -3 microcoulombs). Let's plot the graph for each pair of charges and interpret the results.
Setting Up the Calculation
For the first pair of charges (1 µC and 2 µC):
- q1 = 1 µC = 1 x 10^-6 C
- q2 = 2 µC = 2 x 10^-6 C
For the second pair of charges (2 µC and -3 µC):
- q1 = 2 µC = 2 x 10^-6 C
- q2 = -3 µC = -3 x 10^-6 C
Calculating the Coulomb Force
Now, let's calculate the force for different values of r (the distance between the charges). We'll take values like 0.01 m, 0.02 m, 0.03 m, and so forth, up to 0.1 m. The forces will be calculated using the formula provided above.
Force Calculation for 1 µC and 2 µC
Using the equation:
F1 = k * (|1 x 10^-6 * 2 x 10^-6| / r^2)
This will give us values for F1 at different distances r. For example:
- At r = 0.01 m: F1 = k * (2 x 10^-12 / (0.01)^2) = 1.79 x 10^3 N
- At r = 0.02 m: F1 = k * (2 x 10^-12 / (0.02)^2) = 447.5 N
- At r = 0.03 m: F1 = k * (2 x 10^-12 / (0.03)^2) = 197.2 N
Force Calculation for 2 µC and -3 µC
Now, for the second pair:
F2 = k * (|2 x 10^-6 * -3 x 10^-6| / r^2)
- At r = 0.01 m: F2 = k * (6 x 10^-12 / (0.01)^2) = 5.39 x 10^3 N
- At r = 0.02 m: F2 = k * (6 x 10^-12 / (0.02)^2) = 1347.5 N
- At r = 0.03 m: F2 = k * (6 x 10^-12 / (0.03)^2) = 596.5 N
Graphing the Results
Once we have calculated the forces for both pairs across a range of distances, we can plot these values. The x-axis will represent the distance (r), while the y-axis will represent the Coulomb force (F).
In both graphs, as r increases, the value of F decreases sharply. This is because the force is inversely proportional to the square of the distance. However, due to the differing magnitudes of the charges, the graph for the pair (2 µC and -3 µC) will show a larger force overall compared to the pair (1 µC and 2 µC) at the same distances. This indicates that while both pairs exhibit the same type of relationship, the strength of the force varies significantly based on the charge magnitudes.
Interpreting the Graphs
The graphs will show a hyperbolic decay, characteristic of inverse square relationships:
- The greater the distance, the weaker the force.
- For charges of opposite signs (like 2 µC and -3 µC), the force is attractive, while for like charges (1 µC and 2 µC), the force is repulsive.
In summary, both graphs will illustrate how Coulomb's force diminishes with increasing distance, yet they will also highlight the effect of charge magnitude and sign on the force's strength and nature. Understanding these principles can greatly enhance your grasp of electrostatic interactions in physics.
